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Fel's Conjecture on Syzygies of Numerical Semigroups

Evan Chen, Chris Cummins, GSM, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Markosyan, Ken Ono, Manooshree Patel, Gaurang Pendharkar, Vedant Rathi, Alex Schneidman, Volker Seeker, Shubho Sengupta, Ishan Sinha, Jimmy Xin, Jujian Zhang

TL;DR

The paper proves Fel's conjecture, which expresses the alternating sum of syzygy degrees K_p(S) for a numerical semigroup S in terms of gap power sums G_r(S) and universal symmetric polynomials T_n evaluated at the generator power sums σ_k and δ_k. The authors rewrite the problem using exponential generating functions, isolate the universal identities for T_n, and derive a closed-form formula: K_p(S) = ∑_{r=0}^p binom(p,r) T_{p-r}(σ) G_r(S) + (2^{p+1}/(p+1)) T_{p+1}(δ). The result is fully formalized in Lean/Mathlib via the AxiomProver pipeline, illustrating end-to-end automated proof of a nontrivial conjecture in algebraic combinatorics of numerical semigroups. By connecting Fel’s polynomials to broader contexts (restricted partitions, zig-zag/ Ramanujan q-series), the work highlights deep universal structures governing syzygies and their generating functions with potential implications for toric and combinatorial algebra.

Abstract

Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for $K_p(S)$, for all $p\ge 0$, in terms of the gap power sums $G_r(S)=\sum_{g\notin S} g^r$ and universal symmetric polynomials $T_n$ evaluated at the generator power sums $σ_k=\sum_i d_i^k$ (and $δ_k=(σ_k-1)/2^k$). We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for $T_n$ needed for the derivation. The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture.

Fel's Conjecture on Syzygies of Numerical Semigroups

TL;DR

The paper proves Fel's conjecture, which expresses the alternating sum of syzygy degrees K_p(S) for a numerical semigroup S in terms of gap power sums G_r(S) and universal symmetric polynomials T_n evaluated at the generator power sums σ_k and δ_k. The authors rewrite the problem using exponential generating functions, isolate the universal identities for T_n, and derive a closed-form formula: K_p(S) = ∑_{r=0}^p binom(p,r) T_{p-r}(σ) G_r(S) + (2^{p+1}/(p+1)) T_{p+1}(δ). The result is fully formalized in Lean/Mathlib via the AxiomProver pipeline, illustrating end-to-end automated proof of a nontrivial conjecture in algebraic combinatorics of numerical semigroups. By connecting Fel’s polynomials to broader contexts (restricted partitions, zig-zag/ Ramanujan q-series), the work highlights deep universal structures governing syzygies and their generating functions with potential implications for toric and combinatorial algebra.

Abstract

Let be a numerical semigroup and its semigroup ring. The Hilbert numerator of determines normalized alternating syzygy power sums encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for , for all , in terms of the gap power sums and universal symmetric polynomials evaluated at the generator power sums (and ). We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for needed for the derivation. The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture.
Paper Structure (20 sections, 14 theorems, 75 equations)

This paper contains 20 sections, 14 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Hilbert series of numerical semigroups, and alternating syzygy
  3. Fel's sequence $T_n$ of universal symmetric polynomials
  4. The reappearance of $T_n$ in relation to Ramanujan's $U_{2n}(q)$ and other contexts
  5. Restricted partitions
  6. Recursion via zig-zag numbers
  7. Ramanujan's $q$-series
  8. Main result
  9. Acknowledgments
  10. Background
  11. Syzygy degrees
  12. Universal symmetric polynomials, in terms of $x_i$
  13. Universal symmetric polynomials, in terms of $\sigma_k$
  14. Universal symmetric polynomials, in terms of $\delta_k$
  15. Worked examples
  16. ...and 5 more sections

Key Result

Theorem 1.4

We have $\mathbb C_0(S) = 1$, $\mathbb C_r(S) = 0$ for $1 \le r \le m-2$, and

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: ref:fel0
  • Theorem 1.5: ref:fel
  • Conjecture 1.6: ref:fel
  • Theorem 1.7: ref:felpoly or ref:quasimodular
  • Theorem 1.8: ref:felpoly or ref:quasimodular
  • Proposition 1.9: ref:quasimodular
  • Theorem 1.10: ref:quasimodular
  • ...and 21 more